Second-order approximate reflection coefficients of vertical transversely isotropic thin beds

Abstract

Due to the periodicity and rhythmicity of sedimentation, short-term cycles are commonly developed in sedimentary strata. Under the assumption of a long seismic wavelength, such strata can be regarded as a seismic single thin bed with vertical transversely isotropic (VTI) characteristics. A thin interbed can be formed by the stacking of a series of isotropic and VTI single thin beds. In seismic inversion, the interference of multiples and mode-converted waves generated within and between thin beds and transmission losses are ignored. These interferences are hardly addressed in seismic data processing due to being submerged in first arrivals. In this work, to thin interbeds of isotropic and VTI single thin beds, we propose second-order approximations of Kennett reflection coefficients for PP-, SP-, PS-, and SS-waves, which consider the internal and interlayer wave propagation effects. The numerical analyses show that the proposed approximations are of high accuracy when the P-wave impedance difference in the VTI single thin bed is from − 40 to 100% and of strong anisotropy. The proposed approximations can be used for the efficient and accurate simulation of the wavefields of media with thin interbedding, bringing great potential for the studies of the inversion methods for the internal property parameters of thin interbeds.

Appendices

Appendix 1: Kennett equations and the second-order approximation for isotropic media

Fig. 13

Multilayered media

As shown in Fig. 13, for an isotropic layered model with m interfaces, Kennett (1983) proposed the equations of reflectivity method for calculating the total reflection coefficients at Interface n:

$${\hat{\mathbf{R}}}_{{\text{D}}}^{(n)} = {\mathbf{R}}_{{\text{D}}}^{(n)} + {\mathbf{T}}_{{\text{U}}}^{(n)} {\dot{\mathbf{R}}}_{{\text{D}}}^{(n - 1)} [{\mathbf{I}} - {\mathbf{R}}_{{\text{U}}}^{(n)} {\dot{\mathbf{R}}}_{{\text{D}}}^{(n - 1)} ]^{ - 1} {\mathbf{T}}_{{\text{D}}}^{(n)} , \, n \in [2,m],$$

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where ${\mathbf{R}}_{{\text{D}}}^{{{(}n{)}}}$ and ${\mathbf{T}}_{{\text{D}}}^{{{(}n{)}}}$ are the down-going single-interface reflection and transmission coefficient matrices at Interface n, ${\mathbf{R}}_{{\text{U}}}^{{{(}n{)}}}$ and ${\mathbf{T}}_{{\text{U}}}^{{{(}n{)}}}$ are the up-going coefficient matrices at Interface n, and I is the identity matrix.

The single-interface reflection and transmission coefficient matrices are:

$$\begin{gathered} {\mathbf{R}}_{{\text{D}}} = \left[ {\begin{array}{*{20}c} {r_{{{\text{PP}}}} } & {r_{{{\text{SP}}}} } \\ {r_{{{\text{PS}}}} } & {r_{{{\text{SS}}}} } \\ \end{array} } \right]_{{\text{D}}}^{(n)} ,{\mathbf{R}}_{{\text{U}}} = \left[ {\begin{array}{*{20}c} {r_{{{\text{PP}}}} } & {r_{{{\text{SP}}}} } \\ {r_{{{\text{PS}}}} } & {r_{{{\text{SS}}}} } \\ \end{array} } \right]_{{\text{U}}}^{(n)} , \hfill \\ \, {\mathbf{T}}_{{\text{D}}} = \left[ {\begin{array}{*{20}c} {t_{{{\text{PP}}}} } & {t_{{{\text{SP}}}} } \\ {t_{{{\text{PS}}}} } & {t_{{{\text{SS}}}} } \\ \end{array} } \right]_{{\text{D}}}^{(n)} , \, {\mathbf{T}}_{{\text{U}}} = \left[ {\begin{array}{*{20}c} {t_{{{\text{PP}}}} } & {t_{{{\text{SP}}}} } \\ {t_{{{\text{PS}}}} } & {t_{{{\text{SS}}}} } \\ \end{array} } \right]_{{\text{U}}}^{(n)} , \hfill \\ \end{gathered}$$

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where r and t are the reflection and transmission coefficients of isotropic media, respectively. ${\dot{\mathbf{R}}}_{{\text{D}}}^{(n - 1)}$ is obtained from the total reflection matrix ${\hat{\mathbf{R}}}_{{\text{D}}}^{(n - 1)}$ at Interface n − 1 by the phase shift to the bottom of Interface n:

$${\dot{\mathbf{R}}}_{{\text{D}}}^{(n - 1)} = {\mathbf{E}}^{(n)} {\hat{\mathbf{R}}}_{{\text{D}}}^{(n - 1)} {\mathbf{E}}^{(n)} ,$$

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where E is the phase-shift factor, and its expansion form is shown in Eq. (9).

Yang and Lu (2020) used Taylor’s expansion approach to expand the Kennett equation as:

$${\mathbf{\hat{R}}}_{{\text{D}}}^{{{\text{(}}n)}} = {\mathbf{R}}_{{\text{D}}}^{{{\text{(}}n)}} + {\mathbf{T}}_{{\text{U}}}^{{{\text{(}}n)}} {\mathbf{\dot{R}}}_{{\text{D}}}^{{{\text{(}}n - 1)}} \left[ {{\mathbf{I}} + \sum\limits_{{x = 1}}^{\infty } {({\mathbf{R}}_{{\text{U}}}^{{{\text{(}}n)}} {\mathbf{\dot{R}}}_{{\text{D}}}^{{{\text{(}}n - 1)}} )^{x} } } \right]^{{ - 1}} {\mathbf{T}}_{{\text{D}}}^{{{\text{(}}n)}} ,{\text{ }}n \in [2,m],$$

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Then, they gave the second-order approximations of isotropic thin interbeds:

$${\hat{\mathbf{R}}}_{{\text{D}}}^{{{(}n)}} = {\mathbf{R}}_{{\text{D}}}^{{{(}n)}} + {\mathbf{T}}_{{\text{U}}}^{{{(}n)}} {\dot{\mathbf{R}}}_{{\text{D}}}^{{{(}n - 1)}} [{\mathbf{I}} + {\mathbf{R}}_{{\text{U}}}^{{{(}n)}} {\dot{\mathbf{R}}}_{{\text{D}}}^{{{(}n - 1)}} + ({\mathbf{R}}_{{\text{U}}}^{{{(}n)}} {\dot{\mathbf{R}}}_{{\text{D}}}^{{{(}n - 1)}} )^{2} ]{\mathbf{T}}_{{\text{D}}}^{{{(}n)}} , \, n \in [2,m].$$

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Appendix 2: Graebner equations

Graebner (1992) derived the single-interface reflection and transmission coefficients of VTI layered media:

$${\mathbf{SX}} = {\mathbf{B}},$$

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whose expansion form is:

$$\begin{array}{*{20}c} {\left[ {\begin{array}{*{20}c} {l_{\alpha 1} } & {m_{\beta 1} } & { - l_{\alpha 2} } & { - m_{\beta 2} } \\ {g_{1} } & {h_{1} } & { - g_{2} } & { - h_{2} } \\ {m_{\alpha 1} } & { - l_{\beta 1} } & {m_{\alpha 2} } & { - l_{\beta 2} } \\ {e_{1} } & {f_{1} } & {e_{2} } & {f_{2} } \\ \end{array} } \right]} & \cdot & {\left[ {\begin{array}{*{20}c} {r_{{{\text{pp}}}}^{{\text{D}}} } & {r_{{{\text{sp}}}}^{{\text{D}}} } & {t_{{{\text{pp}}}}^{{\text{U}}} } & {t_{{{\text{sp}}}}^{{\text{U}}} } \\ {r_{{{\text{ps}}}}^{{\text{D}}} } & {r_{{{\text{ss}}}}^{{\text{D}}} } & {t_{{{\text{ps}}}}^{{\text{U}}} } & {t_{{{\text{ss}}}}^{{\text{U}}} } \\ {t_{{{\text{pp}}}}^{{\text{D}}} } & {t_{{{\text{sp}}}}^{{\text{D}}} } & {r_{{{\text{pp}}}}^{{\text{U}}} } & {r_{{{\text{ps}}}}^{{\text{U}}} } \\ {t_{{{\text{ps}}}}^{{\text{D}}} } & {t_{{{\text{ss}}}}^{{\text{D}}} } & {r_{{{\text{sp}}}}^{{\text{U}}} } & {r_{{{\text{ss}}}}^{{\text{U}}} } \\ \end{array} } \right]} & = & {\left[ {\begin{array}{*{20}c} { - l_{\alpha 1} } & { - m_{\beta 1} } & {l_{\alpha 2} } & {m_{\beta 2} } \\ { - g_{1} } & { - h_{1} } & {g_{2} } & {h_{2} } \\ {m_{\alpha 1} } & { - l_{\beta 1} } & {m_{\alpha 2} } & { - l_{\beta 2} } \\ {e_{1} } & {f_{1} } & {e_{2} } & {f_{2} } \\ \end{array} } \right],} \\ \end{array}$$

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where r and t are the reflection and transmission coefficients for a single interface of an VTI medium, respectively, and the superscripts D and U refer to the down-going and up-going waves, respectively.

The following expressions are the direction cosines of the polarization vectors:

$$\begin{gathered} l_{\alpha } = \sqrt {\frac{{a_{33} q_{\alpha }^{2} + a_{55} p^{2} - 1}}{{a_{11} p^{2} + a_{55} q_{\alpha }^{2} - 1 + a_{33} q_{\alpha }^{2} + a_{55} p^{2} - 1}}} , \\ l_{\beta } = \sqrt {\frac{{a_{11} p^{2} + a_{55} q_{\beta }^{2} - 1}}{{a_{11} p^{2} + a_{55} q_{\beta }^{2} - 1 + a_{33} q_{\beta }^{2} + a_{55} p^{2} - 1}}} , \\ m_{\alpha } = \sqrt {\frac{{a_{11} p^{2} + a_{55} q_{\beta }^{2} - 1}}{{a_{11} p^{2} + a_{55} q_{\alpha }^{2} - 1 + a_{33} q_{\alpha }^{2} + a_{55} p^{2} - 1}}} , \\ m_{\beta } = \sqrt {\frac{{a_{33} q_{\beta }^{2} + a_{55} p^{2} - 1}}{{a_{11} p^{2} + a_{55} q_{\beta }^{2} - 1 + a_{33} q_{\beta }^{2} + a_{55} p^{2} - 1}}} , \\ \end{gathered}$$

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with a_ij≡c_ij/ρ, where c_ij is the stiffness component, ρ is the density of the layer, q_α and q_β are the vertical slowness of P- and S-waves, respectively; and p is the horizontal slowness. Subscripts 1 and 2 refer to the lower and upper layers, respectively.

$$\begin{gathered} e = a_{55} (q_{\alpha }^{{}} l_{\alpha } + pm_{\alpha } ), \\ f = a_{55} (q_{\beta }^{{}} m_{\beta } + pl_{\beta } ), \\ g = pl_{\alpha } a_{13} + q_{\alpha }^{{}} m_{\alpha } a_{33} , \\ h = pm_{\beta } a_{13} - q_{\beta }^{{}} l_{\beta } a_{33} . \\ \end{gathered}$$

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